SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Anonymous and Copy-Robust Delegations for Liquid Democracy*
Markus Utke
TU Eindhoven
Eindhoven, The Netherlands
m.utke@tue.nl
Ulrike Schmidt-Kraepelin
Simons Laufer Mathematical Sciences
Institute (SLMath)
Berkeley, CA, United States
uschmidt@slmath.org
Liquid democracy with ranked delegations is a novel voting scheme that unites the practicability of representative democracy with the idealistic appeal of direct democracy: Every voter decides between casting their vote on a question at hand or delegating their voting weight to some other, trusted agent. Delegations are transitive, and since voters may end up in a delegation cycle, they are encouraged to indicate not only a single delegate, but a set of potential delegates and a ranking among them. Based on the delegation preferences of all voters, a delegation rule selects one representative per voter. Previous work has revealed a trade-off between two properties of delegation rules called anonymity and copy-robustness.
To overcome this issue we study two fractional delegation rules: MIXED BORDA BRANCHING, which generalizes a rule satisfying copy-robustness, and the RANDOM WALK RULE, which satisfies anonymity. Using the Markov chain tree theorem, we show that the two rules are in fact equivalent, and simultaneously satisfy generalized versions of the two properties. Combining the same theorem with Fulkerson's algorithm, we develop a polynomial-time algorithm for computing the outcome of the studied delegation rule. This algorithm is of independent interest, having applications in semi-supervised learning and graph theory.
Today, democratic decision-making in legislative bodies, parties, and non-profit organizations is often done via one of two extremes: In representative democracy, the constituents elect representatives who are responsible for deciding upon all upcoming issues for a period of several years. In direct democracy, the voters may vote upon every issue themselves. While the latter is distinguished by its idealistic character, it may suffer from low voter turnout as voters do not feel sufficiently informed. Liquid democracy aims to provide the best of both worlds by letting voters decide whether they want to cast their opinion on an issue at hand, or prefer to delegate their voting weight to some other, trusted voter. Delegations are transitive, i.e., if voter  $v_{1}$  delegates to voter  $v_{2}$ , and  $v_{2}$  in turn delegates to voter  $v_{3}$ , who casts its vote, then  $v_{3}$  receives the voting weight of both  $v_{1}$  and  $v_{2}$ . Liquid democracy has been implemented, for example, by political parties [Kling et al., 2015] and Google [Hardt and Lopes, 2015]. From a theoretic viewpoint, liquid democracy has been studied intensively by the social choice community in the last decade [Paulin, 2020].
Earlier works on liquid democracy [Christoff and Grossi, 2017, Brill, 2018] have pointed towards the issue of delegation cycles, e.g., the situation that occurs when voter  $v_{3}$  in the above example decides to delegate to voter  $v_{1}$  instead of casting its vote. If this happens, none of the three voters reaches a
*Part of this research was carried out while both authors were affiliated with TU Berlin and Universidad de Chile, and Markus Utke was affiliated with University of Amsterdam.
casting voter via a chain of trusted delegations, and therefore their voting weight would be lost. In order to reduce the risk of the appearance of such so-called isolated voters, several scholars suggested to allow voters to indicate back-up delegations [Brill, 2018, Gölz et al., 2021, Kavitha et al., 2022] that may be used in case there is no delegation chain using only top-choice delegations. In liquid democracy with ranked delegations [Brill et al., 2022, Colley et al., 2022, Kavitha et al., 2022], voters are assumed to indicate a set of trusted delegates together with a ranking (preference order) among them. In fact, Brill et al. [2022] showed empirically that in many random graph models, one to two back-up delegations per voter suffice in order to avoid the existence of isolated voters almost entirely.
Allowing the voters to indicate multiple possible delegations calls for a principled way to decide between multiple possible delegation chains. For example, consider Figure 1: Should voter  $v_{1}$ 's weight be assigned to casting voter  $s_{1}$ , via  $v_{1}$ 's second-ranked delegation, or should it rather be assigned to voter  $s_{2}$  by following  $v_{1}$ 's first-ranked delegation to voter  $v_{2}$ , and then following  $v_{2}$ 's second-ranked delegation? Brill et al. [2022] introduced the concept of delegation rules, which take as input a delegation graph (i.e., a digraph with a rank function on the edges) and output an assignment of each (non-isolated) delegating voter to a casting voter. In order to navigate within the space of delegation rules, they apply the axiomatic method [Thomson, 2001], as commonly used in social choice theory. In particular, the authors argue that the following three axioms are desirable:
(i) Confluence: A delegation rule selects one path from every delegating voter to a casting voter and these paths are "consistent" with one another. That is, when the path of voter  $v_{1}$  reaches some other delegating voter  $v_{2}$ , the remaining subpath of  $v_{1}$  coincides with the path of  $v_{2}$ . This was argued to increase accountability of delegates [Brill et al., 2022, Gölz et al., 2021].
(ii) Anonymity: A delegation rule should make decisions solely on the structure of the graph, not on the identities of the voters, i.e. it should be invariant under renaming of the voters.
(iii) Copy-robustness: If a delegating voter  $v_{1}$  decides to cast her vote herself instead of delegating, this should not change the sum of the voting weight assigned to  $v_{1}$ 's representative and herself. This property was emphasized by practitioners [Behrens and Swierczek, 2015] to avoid manipulations in the system by delegating voters acting as casting voters but actually copying the vote of their former representative.
Figure 1: Example of a delegation graph. Delegating voters  $(v_{1}$  and  $v_{2})$  are indicated by circles and casting voters  $(s_{1}$  and  $s_{2})$  by squares. Solid edges represent the first-ranked delegations and dashed edges second-ranked delegations.
For any pair of axioms (i) to (iii), Brill et al. [2022] provide a delegation rule that satisfies both of them. In contrast, we prove in Section 7 that there exists no delegation rule that satisfies all three properties simultaneously, thereby strengthening an impossibility result by Brill et al. [2022].<sup>2</sup>
Our Contribution We show that the above impossibility is due to the restriction that delegation rules may not distribute the voting weight of a delegating voter to more than one casting voter.
- We generalize the definition of delegation rules to fractional delegation rules (Section 3) and provide generalizations of all three axioms above (Section 7).
- We introduce a natural variant of the BORDA BRANCHING rule [Brill et al., 2022], which we call MIXED BORDA BRANCHING. We show that this rule is equivalent to the RANDOM WALK RULE, a fractional delegation rule that has been suggested by Brill [2018].
- In our main result, we build upon Fulkerson's algorithm [Fulkerson, 1974] and the Markov chain tree theorem [Leighton and Rivest, 1986] and show the existence of a polynomial-time algorithm for MIXED BORDA BRANCHING. This algorithm is of independent interest, as it computes the probability of two nodes being connected, when sampling a min-cost branching in a digraph uniformly at random. This problem features in semi-supervised learning, under the name directed power watershed [Fita Sanmartin et al., 2021], a directed
<sup>2</sup>Brill et al. [2022] show that there exists no delegation rule belonging to the subclass of sequence rules that is both confluent and copy-robust. Any sequence rule is in particular anonymous.
variant of the power watershed [Couprie et al., 2010]. To the best of our knowledge, we provide the first efficient algorithm.
- In Section 7, we show that the RANDOM WALK RULE (and thus MIXED BORDA BRANCHING) satisfies the generalizations of all three axioms. We also formalize the impossibility for non-fractional delegation rules. Beyond that, we show that the RANDOM WALK RULE satisfies a generalization of a further axiom (guru participation) which has been studied in the literature [Kotsialou and Riley, 2020, Colley et al., 2022, Brill et al., 2022] (Appendix C).
The proofs (or their completions) for results marked by  $(\star)$  can be found in the appendix.
Related Work Liquid democracy. The idea to let agents rank potential delegates in liquid democracy was first presented by the developers of the liquid democracy platform Liquid Feedback [Behrens and Swierczek, 2015], who presented seven properties that cannot be satisfied simultaneously. Some of these properties, such as copy-robustness and guru participation, have been picked up in the social choice literature [Kotsialou and Riley, 2020, Brill et al., 2022, Colley et al., 2022]. The connection of confluent delegation rules to branchings in a digraph was first emphasized by Kavitha et al. [2022] and later built upon in [Brill et al., 2022, Natsui and Takazawa, 2022]. We base our model on [Brill et al., 2022], as their model captures all rules and axioms from the literature. Fractional delegations were studied by Degrave [2014] and Bersetche [2022], however, here agents indicate a desired distribution among their delegates instead of a ranking. While the two approaches are orthogonal, we argue in Appendix B that they could be easily combined (and our algorithm could be adjusted).
Branchings and matrix tree theorems. Our algorithm for computing MIXED BORDA BRANCHING is based on an algorithm for computing min-cost branchings in directed trees. This can be done, e.g., via Edmond's algorithm [Edmonds, 1967] or Fulkerson's algorithm [Fulkerson, 1974]. The latter comes with a characterization of min-cost branchings in terms of dual certificates, which we utilize in Algorithm 2. We refer to Kamiyama [2014] for a comprehensive overview on the literature of min-cost branchings. Our algorithm makes use of (a directed version of) the matrix tree theorem [Tutte, 1948], which allows to count directed trees in a digraph. An extension of this theorem is the Markov chain tree theorem [Leighton and Rivest, 1986], which we use for the construction of our algorithm as well as for proving the equivalence of MIXED BORDA BRANCHING and the RANDOM WALK RULE. A comprehensive overview of the literature is given by Pitman and Tang [2018].
Semi-supervised learning. There is a connection of our setting to graph-based semi-supervised learning. In particular, Algorithm 2 is related to the power watershed algorithm [Couprie et al., 2010] and the probabilistic watershed algorithm [Fita Sanmartin et al., 2019, 2021]. We elaborate on this connection in Section 4.
The main mathematical concepts used in this paper are directed graphs (also called digraphs), branchings, in-trees, and Markov chains, all of which we briefly introduce below.
We assume that a digraph  $G$  has no parallel edges, and denote by  $V(G)$  the set of nodes and by  $E(G)$  the set of edges of  $G$ . We use  $\delta_G^+(v)$  to indicate the set of outgoing edges of node  $v \in V(G)$ , i.e.,  $\delta_G^+(v) = \{(v, u) \in E(G)\}$ . For a set of nodes  $U \subseteq V(G)$ , we define the outgoing cut of  $U$  by  $\delta_G^+(U) = \{(u, v) \in E(G) \mid u \in U, v \in V(G) \setminus U\}$ . A walk  $W$  is a node sequence  $(W_1, \ldots, W_{|W|})$  such that  $(W_i, W_{i+1}) \in E(G)$  for  $i \in \{1, \ldots, |W| - 1\}$ . We omit  $G$  if it is clear from the context.
Branchings and in-trees. Given a digraph  $G$ , we say that  $B \subseteq E$  is a branching (or in-forest) in  $G$ , if  $B$  is acyclic, and the out-degree of any node  $v \in V(G)$  in  $B$  is at most one, i.e.,  $|B \cap \delta^{+}(v)| \leq 1$ . Throughout the paper, we use the term branching to refer to maximum-cardinality branchings, i.e., branchings  $B$  maximizing  $|B|$  among all branchings in  $G$ . For a given digraph  $G$  we define  $\mathcal{B}(G)$  as the set of all (max-cardinality) branchings and  $\mathcal{B}_{v,s}(G)$  as the set of all (max-cardinality) branchings in which node  $v \in V(G)$  has a path to the node  $s \in V(G)$ . For any branching  $B$  in any digraph  $G$  it holds that  $|B| \leq |V(G)| - 1$ . If in fact  $|B| = |V(G)| - 1$ , then  $B$  is also called an in-tree. For every in-tree  $T \subseteq E$ , there exists exactly one node  $v \in V(G)$  without outgoing edge. In this case, we also say that  $v$  is the sink of  $T$  and call  $T$  a  $v$ -tree. For  $v \in V(G)$ , we let  $\mathcal{T}_v(G)$  be the set of  $v$ -trees in  $G$ .
Matrix tree theorem. For our main result we need to count the number of in-trees in a weighted digraph, which can be done with help of the matrix tree theorem. For a digraph  $G$  with weight
function  $w: E \to \mathbb{N}$ , we define the weight of a subgraph  $T \subseteq E$  as  $w(T) = \prod_{e \in T} w(e)$  and the weight of a collection of subgraphs  $\mathcal{T}$  as  $w(\mathcal{T}) = \sum_{T \in \mathcal{T}} w(T)$ . Then, we define the Laplacian matrix of  $(G, w)$  as  $L = D - A$ , where  $D$  is a diagonal matrix containing the weighted out-degree of any node  $v$  in the corresponding entry  $D_{v,v}$  and  $A$  is the weighted adjacency matrix, given as  $A_{u,v} = w((u,v))$  for any edge  $(u,v) \in E$  and zero everywhere else. Moreover, we denote by  $L^{(v)}$  the matrix resulting from  $L$  when deleting the row and column corresponding to  $v$ .
Lemma 1 (Matrix tree theorem [Tutte, 1948, De Leenheer, 2020]). Let  $(G,w)$  be a weighted digraph and let  $L$  be its Laplacian matrix. Then,
$$
d e t (L ^ {(v)}) = \sum_ {T \in \mathcal {T} _ {v}} \prod_ {e \in T} w (e) = w (\mathcal {T} _ {v}).
$$
If we interpret the weight of an edge as its multiplicity in a multigraph, then  $\operatorname{det}(L^{(v)})$  equals the total number of distinct  $v$ -trees.
Markov Chains. A Markov chain is a tuple  $(G, P)$ , where  $G$  is a digraph and the matrix  $P \in [0,1]^{|V| \times |V|}$  encodes the transition probabilities. That is, the entry  $P_{u,v}$  indicates the probability with which the Markov chain moves from state  $u$  to state  $v$  in one timestep. For a given edge  $e = (u,v) \in E$ , we also write  $P_e$  to refer to  $P_{u,v}$ . For all  $v \in V$  it holds that  $\sum_{e \in \delta^+(v)} P_e = 1$ . Moreover, if  $(u,v) \notin E$ , we assume  $P_{u,v} = 0$ . We define the matrix  $Q \in [0,1]^{|V| \times |V|}$  as  $Q = \lim_{\tau \to \infty} \frac{1}{\tau} \sum_{i=0}^{\tau} P^{\tau}$ . If  $(G,P)$  is an absorbing Markov chain,  $Q_{u,v}$  corresponds to the probability for a random walk starting in  $u$  to end in absorbing state  $v$ . In contrast, if  $G$  is strongly connected and  $P$  is positive for all edges in  $G$ , then  $Q_{u,v}$  corresponds to the relative number of times  $v$  is visited in an infinite random walk independent of the starting state [Grinstead and Snell, 1997].
3 Liquid Democracy with Fractional Delegations
A delegation graph  $G = (N \cup S, E)$  is a digraph with a cost function  $c: E \to \mathbb{N}$  (called rank function before), representing the preferences of the voters over their potential delegates (lower numbers are preferred). Nodes correspond to voters and an edge  $(u, v) \in E$  indicates that  $u$  accepts  $v$  as a delegate. By convention, the set of nodes  $S$  corresponds exactly to the sinks of the digraph, i.e., the set of nodes without outgoing edges. Thus,  $S$  captures the casting voters, and  $N$  the delegating voters. We assume for all  $v \in N$  that they reach some element in  $S$ . A delegation rule maps any delegation graph to a fractional assignment, i.e., a matrix  $A \in [0,1]^{N \times S}$ , where, for every  $v \in N$ ,  $s \in S$ , the entry  $A_{v,s}$  indicates the fraction of delegating voter  $v$ 's weight that is allocated to casting voter  $s \in S$ . For any  $v \in N$  we refer to any casting voter  $s \in S$  with  $A_{v,s} > 0$  as a representative of  $v$ . For assignment  $A$ , the voting weight of a casting voter  $s \in S$  is defined as  $\pi_s(A) = 1 + \sum_{v \in N} A_{v,s}$ . A non-fractional delegation rule is a special case of a delegation rule, that always returns assignments  $A \in \{0,1\}^{N \times S}$ .
MIXED BORDA BRANCHING. The output of any non-fractional, confluent delegation rule can be represented as a branching: Any branching in the delegation graph consists of  $|N|$  edges, and every delegating voter has a unique path to some casting voter. Brill et al. [2022] suggested to select min-cost branchings, i.e., those minimizing  $\sum_{e \in B} c(e)$ . The authors call these objects Borda branchings and show that selecting them yields a copy-robust rule. As this rule is inherently non-anonymous, we suggest to mix uniformly over all Borda branchings, hoping to gain anonymity without losing the other properties<sup>7</sup>. Formally, for a given delegation graph  $(G, c)$ , let  $\mathcal{B}^*(G)$  be the set of all Borda branchings and let  $\mathcal{B}_{v,s}^*(G)$  be the set of all Borda branchings in which delegating voter  $v \in N$  is connected to casting voter  $s \in S$ . MIXED BORDA BRANCHING returns the assignment  $A$  defined as
$$
A _ {v, s} = \frac {| \mathcal {B} _ {v , s} ^ {*} (G) |}{| \mathcal {B} ^ {*} (G) |} \quad \text {f o r a l l} v \in V, s \in S.
$$
3 Tutte [1948] proves the theorem for digraphs, the weighted version can be found in [De Leenheer, 2020].
<sup>4</sup>We remark that in this paper cost functions and weight functions play different roles.
<sup>5</sup>If this is not the case, such a voter is isolated, i.e., there is no chain of trusted delegations to a casting voter, thus we cannot meaningfully assign its voting weight. We remove isolated nodes in a preprocessing step.
<sup>6</sup>While this definition allows  $A_{v,s} > 0$  for any  $v \in N, s \in S$ , for a sensible delegation rule, this should only be the case if there exists a path from  $v$  to  $s$ . Confluence (defined in Section 7) implies this restriction.
7Confluence and copy-robustness are not directly inherited from the non-fractional counterpart of the rule.
RANDOM WALK RULE. The second delegation rule was suggested in Brill [2018] and attributed to Vincent Conitzer. For any given delegation graph  $(G, c)$  and fixed  $\varepsilon \in (0, 1]$ , we define a Markov chain on  $G$ , where the transition probability matrix  $P^{(\varepsilon)} \in [0, 1]^{V(G) \times V(G)}$  is defined as
$$
P _ {u, v} ^ {(\varepsilon)} = \frac {\varepsilon^ {c (u , v)}}{\bar {\varepsilon} _ {u}} \quad \text {f o r a n y} (u, v) \in E, \tag {1}
$$
where  $\bar{\varepsilon}_u = \sum_{(u,v)\in \delta^+ (u)}\varepsilon^{c(u,v)}$  is the natural normalization factor. The Markov chain  $(G,P^{(\varepsilon)})$  is absorbing for every  $\varepsilon \in (0,1]$  where the absorbing states are exactly  $S$ . The RANDOM WALK RULE returns the fractional assignment  $A$  defined as the limit of the absorbing probabilities, i.e.,
$$
A _ {v, s} = \lim  _ {\varepsilon \rightarrow 0} \left(\lim  _ {\tau \rightarrow \infty} \frac {1}{\tau} \sum_ {i = 0} ^ {\tau} (P ^ {(\varepsilon)}) ^ {\tau}\right) _ {v, s} \quad \text {f o r a l l} v \in N, s \in S.
$$
4 Connection to Semi-Supervised Learning
In graph-based semi-supervised learning, the input is a directed or undirected graph, where nodes correspond to data points and edges correspond to relationships between these. A subset of the nodes is labeled and their labels are used to classify unlabeled data. Many algorithms compute a fractional assignment of unlabeled data points towards labeled data points, which is then used to determine the predictions for unlabeled data. In the directed case, there exists a one-to-one correspondence to our model: Delegating voters correspond to unlabeled data and casting voters correspond to labeled data.
Power Watershed. In the undirected case, the power watershed [Couprie et al., 2010] can be interpreted as an undirected analog of MIXED BORDA BRANCHING: For any pair of unlabeled data point  $x$  and labeled data point  $y$ , the algorithm computes the fraction of min-cost undirected maximum forests that connect  $x$  to  $y$ . The authors provide a polynomial-time algorithm for computing its outcome. On a high level, our algorithm (Section 5) is reminiscent of theirs, i.e., both algorithms iteratively contract parts of the graph, leading to a hierarchy of subsets of the nodes. However, while the algorithm by Couprie et al. [2010] only needs to carry out computations at the upper level of the hierarchy, our algorithm has to carry out computations at each level of the hierarchy. We believe that the increased complexity of our algorithm is inherent to our problem and do not find this surprising: Even the classic min-cost spanning tree problem can be solved by a greedy algorithm and forms a Matroid, but this structure gets lost when moving to its directed variant.
Directed Probabilistic Watershed. Fita Sanmartin et al. [2021] study a directed version of graph-based semi-supervised learning. The authors introduce the directed probabilistic watershed (DProbWS): Similar to our model, there exists a cost function  $c$  and a weight function  $w$  on the edges. In their case, these functions are linked by  $w(e) = \exp(-\mu c(e))$ . The paper studies a Gibbs distribution with respect to the weights, i.e., the probability of sampling a branching is proportional to its weight. The parameter  $\mu$  controls the entropy of this function, i.e., for  $\mu = 0$ , any branching is sampled with equal probability, while larger  $\mu$  places more probability on low cost branchings. The authors show: (i) For fixed  $\mu$ , the fractional allocation induced by the Gibbs distribution can be computed by calculating the absorbing probabilities of a Markov chain. (ii) For the limit case  $\mu \to \infty$ , the defined distribution equals the uniform distribution over all min-cost branchings, i.e., the distribution that we study in this paper. In this limit case, the authors refer to the corresponding solution as the directed power watershed. Hence, the authors have shown the equivalence of the directed power watershed and the limit of a parameterized Markov chain. Since this Markov chain only slightly differs from ours, this result is very close to our Theorem 5. As its proof is very short and might be of interest for the reader, we still present it in Section 6. Importantly, Fita Sanmartin et al. [2021] do not show how to compute the outcome of the directed power watershed. In particular, the algorithm from (i) makes explicit use of the weight function on the edges (which depends on  $\mu$ ). Hence, the running time of the algorithm grows to infinity in the limit case. We fill this gap by providing the (to the best of our knowledge) first polynomial-time algorithm for the directed power watershed. Maybe surprisingly, we need a significantly more complex approach to solve the limit case: While the algorithm of Fita Sanmartin et al. [2021] solves one absorbing Markov chain, our algorithm derives a hierarchical structure of subsets of the nodes by the structural insights provided by Fulkerson's algorithm, and solves several Markov chains for each of the hierarchy.
This analogy holds for the variant of the power watershed when a parameter  $q$  is 2 [Couprie et al., 2010].
5 Computation of MIXED BORDA BRANCHING